local field

A local field is a topological field which is Hausdorff and locally compact as a topological space.

Examples of local fields include:

• Any field together with the discrete topology.
• The field $\mathbb{R}$ of real numbers.
• The field $\mathbb{C}$ of complex numbers.
• The field $\mathbb{Q}_p$ of $p$ -adic rationals, or any finite extension thereof.
• The field $\mathbb{F}_q((t))$ of formal Laurent series in one variable $t$ with coefficients in the finite field $\mathbb{F}_q$ of $q$ elements.

In fact, this list is complete--every local field is isomorphic as a topological field to one of the above fields.

Acknowledgements

This document is dedicated to those who made it all the way through Serre's book [1] before realizing that nowhere within the book is there a definition of the term ``local field.''

Bibliography

1

Jean-Pierre Serre, Local Fields, Springer-Verlag, 1979 (GTM 67).